Calculation:

z = ((-5i)^(1/8))*(8^(1/3))

Result:



There are 24 solutions, due to “The Fundamental Theorem of Algebra”. Your expression contains square roots or powers to 1/n.

z1 = ((-5i)^(1/8))*(8^(1/3)) = 2.39869586-0.47713027i = 2.44568909 × ei (-π/16) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei (-π/16) = 1.19934793-0.23856514i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: (1.19934793-0.23856514i) * 2 = 2.39869586-0.47713027i

z2 = ((-5i)^(1/8))*(8^(1/3)) = 2.03351616+1.35875206i = 2.44568909 × ei 3π/16 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei 3π/16 = 1.01675808+0.67937603i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: (1.01675808+0.67937603i) * 2 = 2.03351616+1.35875206i

z3 = ((-5i)^(1/8))*(8^(1/3)) = 0.47713027+2.39869586i = 2.44568909 × ei 7π/16 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei 7π/16 = 0.23856514+1.19934793i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: (0.23856514+1.19934793i) * 2 = 0.47713027+2.39869586i

z4 = ((-5i)^(1/8))*(8^(1/3)) = -1.35875206+2.03351616i = 2.44568909 × ei 11π/16 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei 11π/16 = -0.67937603+1.01675808i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: (-0.67937603+1.01675808i) * 2 = -1.35875206+2.03351616i

z5 = ((-5i)^(1/8))*(8^(1/3)) = -2.39869586+0.47713027i = 2.44568909 × ei 15π/16 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei 15π/16 = -1.19934793+0.23856514i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: (-1.19934793+0.23856514i) * 2 = -2.39869586+0.47713027i

z6 = ((-5i)^(1/8))*(8^(1/3)) = -2.03351616-1.35875206i = 2.44568909 × ei (-13π/16) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei (-13π/16) = -1.01675808-0.67937603i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: (-1.01675808-0.67937603i) * 2 = -2.03351616-1.35875206i

z7 = ((-5i)^(1/8))*(8^(1/3)) = -0.47713027-2.39869586i = 2.44568909 × ei (-9π/16) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei (-9π/16) = -0.23856514-1.19934793i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: (-0.23856514-1.19934793i) * 2 = -0.47713027-2.39869586i

z8 = ((-5i)^(1/8))*(8^(1/3)) = 1.35875206-2.03351616i = 2.44568909 × ei (-5π/16) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei (-5π/16) = 0.67937603-1.01675808i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = 2
  5. Multiple: (0.67937603-1.01675808i) * 2 = 1.35875206-2.03351616i

z9 = ((-5i)^(1/8))*(8^(1/3)) = -0.78614099+2.31589669i = 2.44568909 × ei 29π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei (-π/16) = 1.19934793-0.23856514i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: (1.19934793-0.23856514i) * (-1+1.73205081i) = 1.1993479299514 * (-1) + 1.1993479299514 * 1.7320508075689i + (-0.23856513605852i) * (-1) + (-0.23856513605852i) * 1.7320508075689i = -1.19934793+2.07733155i+0.23856514i-0.41320694i2 = -1.19934793+2.07733155i+0.23856514i+0.41320694 = -1.19934793 + 0.41320694 +i(2.07733155 + 0.23856514) = -0.78614099+2.31589669i
    alternative steps
    1.22284454 × ei (-π/16) × 2 × ei 2π/3 = 1.2228445449939 × 2 × ei ((-π/16)+2π/3) = 2.44568909 × ei 29π/48 = -0.78614099+2.31589669i

z10 = ((-5i)^(1/8))*(8^(1/3)) = -2.19347188+1.08170062i = 2.44568909 × ei 41π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei 3π/16 = 1.01675808+0.67937603i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: (1.01675808+0.67937603i) * (-1+1.73205081i) = 1.0167580797323 * (-1) + 1.0167580797323 * 1.7320508075689i + 0.67937602880898i * (-1) + 0.67937602880898i * 1.7320508075689i = -1.01675808+1.76107665i-0.67937603i+1.1767138i2 = -1.01675808+1.76107665i-0.67937603i-1.1767138 = -1.01675808 - 1.1767138 +i(1.76107665 - 0.67937603) = -2.19347188+1.08170062i
    alternative steps
    1.22284454 × ei 3π/16 × 2 × ei 2π/3 = 1.2228445449939 × 2 × ei (3π/16+2π/3) = 2.44568909 × ei 41π/48 = -2.19347188+1.08170062i

z11 = ((-5i)^(1/8))*(8^(1/3)) = -2.31589669-0.78614099i = 2.44568909 × ei (-43π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei 7π/16 = 0.23856514+1.19934793i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: (0.23856514+1.19934793i) * (-1+1.73205081i) = 0.23856513605852 * (-1) + 0.23856513605852 * 1.7320508075689i + 1.1993479299514i * (-1) + 1.1993479299514i * 1.7320508075689i = -0.23856514+0.41320694i-1.19934793i+2.07733155i2 = -0.23856514+0.41320694i-1.19934793i-2.07733155 = -0.23856514 - 2.07733155 +i(0.41320694 - 1.19934793) = -2.31589669-0.78614099i
    alternative steps
    1.22284454 × ei 7π/16 × 2 × ei 2π/3 = 1.2228445449939 × 2 × ei (7π/16+2π/3) = 2.44568909 × ei (-43π/48) = -2.31589669-0.78614099i

z12 = ((-5i)^(1/8))*(8^(1/3)) = -1.08170062-2.19347188i = 2.44568909 × ei (-31π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei 11π/16 = -0.67937603+1.01675808i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: (-0.67937603+1.01675808i) * (-1+1.73205081i) = -0.67937602880898 * (-1) + (-0.67937602880898) * 1.7320508075689i + 1.0167580797323i * (-1) + 1.0167580797323i * 1.7320508075689i = 0.67937603-1.1767138i-1.01675808i+1.76107665i2 = 0.67937603-1.1767138i-1.01675808i-1.76107665 = 0.67937603 - 1.76107665 +i(-1.1767138 - 1.01675808) = -1.08170062-2.19347188i
    alternative steps
    1.22284454 × ei 11π/16 × 2 × ei 2π/3 = 1.2228445449939 × 2 × ei (11π/16+2π/3) = 2.44568909 × ei (-31π/48) = -1.08170062-2.19347188i

z13 = ((-5i)^(1/8))*(8^(1/3)) = 0.78614099-2.31589669i = 2.44568909 × ei (-19π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei 15π/16 = -1.19934793+0.23856514i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: (-1.19934793+0.23856514i) * (-1+1.73205081i) = -1.1993479299514 * (-1) + (-1.1993479299514) * 1.7320508075689i + 0.23856513605852i * (-1) + 0.23856513605852i * 1.7320508075689i = 1.19934793-2.07733155i-0.23856514i+0.41320694i2 = 1.19934793-2.07733155i-0.23856514i-0.41320694 = 1.19934793 - 0.41320694 +i(-2.07733155 - 0.23856514) = 0.78614099-2.31589669i
    alternative steps
    1.22284454 × ei 15π/16 × 2 × ei 2π/3 = 1.2228445449939 × 2 × ei (15π/16+2π/3) = 2.44568909 × ei (-19π/48) = 0.78614099-2.31589669i

z14 = ((-5i)^(1/8))*(8^(1/3)) = 2.19347188-1.08170062i = 2.44568909 × ei (-7π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei (-13π/16) = -1.01675808-0.67937603i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: (-1.01675808-0.67937603i) * (-1+1.73205081i) = -1.0167580797323 * (-1) + (-1.0167580797323) * 1.7320508075689i + (-0.67937602880898i) * (-1) + (-0.67937602880898i) * 1.7320508075689i = 1.01675808-1.76107665i+0.67937603i-1.1767138i2 = 1.01675808-1.76107665i+0.67937603i+1.1767138 = 1.01675808 + 1.1767138 +i(-1.76107665 + 0.67937603) = 2.19347188-1.08170062i
    alternative steps
    1.22284454 × ei (-13π/16) × 2 × ei 2π/3 = 1.2228445449939 × 2 × ei ((-13π/16)+2π/3) = 2.44568909 × ei (-7π/48) = 2.19347188-1.08170062i

z15 = ((-5i)^(1/8))*(8^(1/3)) = 2.31589669+0.78614099i = 2.44568909 × ei 5π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei (-9π/16) = -0.23856514-1.19934793i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: (-0.23856514-1.19934793i) * (-1+1.73205081i) = -0.23856513605852 * (-1) + (-0.23856513605852) * 1.7320508075689i + (-1.1993479299514i) * (-1) + (-1.1993479299514i) * 1.7320508075689i = 0.23856514-0.41320694i+1.19934793i-2.07733155i2 = 0.23856514-0.41320694i+1.19934793i+2.07733155 = 0.23856514 + 2.07733155 +i(-0.41320694 + 1.19934793) = 2.31589669+0.78614099i
    alternative steps
    1.22284454 × ei (-9π/16) × 2 × ei 2π/3 = 1.2228445449939 × 2 × ei ((-9π/16)+2π/3) = 2.44568909 × ei 5π/48 = 2.31589669+0.78614099i

z16 = ((-5i)^(1/8))*(8^(1/3)) = 1.08170062+2.19347188i = 2.44568909 × ei 17π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei (-5π/16) = 0.67937603-1.01675808i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1+1.73205081i
  5. Multiple: (0.67937603-1.01675808i) * (-1+1.73205081i) = 0.67937602880898 * (-1) + 0.67937602880898 * 1.7320508075689i + (-1.0167580797323i) * (-1) + (-1.0167580797323i) * 1.7320508075689i = -0.67937603+1.1767138i+1.01675808i-1.76107665i2 = -0.67937603+1.1767138i+1.01675808i+1.76107665 = -0.67937603 + 1.76107665 +i(1.1767138 + 1.01675808) = 1.08170062+2.19347188i
    alternative steps
    1.22284454 × ei (-5π/16) × 2 × ei 2π/3 = 1.2228445449939 × 2 × ei ((-5π/16)+2π/3) = 2.44568909 × ei 17π/48 = 1.08170062+2.19347188i

z17 = ((-5i)^(1/8))*(8^(1/3)) = -1.61255487-1.83876641i = 2.44568909 × ei (-35π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei (-π/16) = 1.19934793-0.23856514i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: (1.19934793-0.23856514i) * (-1-1.73205081i) = 1.1993479299514 * (-1) + 1.1993479299514 * (-1.7320508075689i) + (-0.23856513605852i) * (-1) + (-0.23856513605852i) * (-1.7320508075689i) = -1.19934793-2.07733155i+0.23856514i+0.41320694i2 = -1.19934793-2.07733155i+0.23856514i-0.41320694 = -1.19934793 - 0.41320694 +i(-2.07733155 + 0.23856514) = -1.61255487-1.83876641i
    alternative steps
    1.22284454 × ei (-π/16) × 2 × ei (-2π/3) = 1.2228445449939 × 2 × ei ((-π/16)+(-2π/3)) = 2.44568909 × ei (-35π/48) = -1.61255487-1.83876641i

z18 = ((-5i)^(1/8))*(8^(1/3)) = 0.15995572-2.44045268i = 2.44568909 × ei (-23π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei 3π/16 = 1.01675808+0.67937603i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: (1.01675808+0.67937603i) * (-1-1.73205081i) = 1.0167580797323 * (-1) + 1.0167580797323 * (-1.7320508075689i) + 0.67937602880898i * (-1) + 0.67937602880898i * (-1.7320508075689i) = -1.01675808-1.76107665i-0.67937603i-1.1767138i2 = -1.01675808-1.76107665i-0.67937603i+1.1767138 = -1.01675808 + 1.1767138 +i(-1.76107665 - 0.67937603) = 0.15995572-2.44045268i
    alternative steps
    1.22284454 × ei 3π/16 × 2 × ei (-2π/3) = 1.2228445449939 × 2 × ei (3π/16+(-2π/3)) = 2.44568909 × ei (-23π/48) = 0.15995572-2.44045268i

z19 = ((-5i)^(1/8))*(8^(1/3)) = 1.83876641-1.61255487i = 2.44568909 × ei (-11π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei 7π/16 = 0.23856514+1.19934793i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: (0.23856514+1.19934793i) * (-1-1.73205081i) = 0.23856513605852 * (-1) + 0.23856513605852 * (-1.7320508075689i) + 1.1993479299514i * (-1) + 1.1993479299514i * (-1.7320508075689i) = -0.23856514-0.41320694i-1.19934793i-2.07733155i2 = -0.23856514-0.41320694i-1.19934793i+2.07733155 = -0.23856514 + 2.07733155 +i(-0.41320694 - 1.19934793) = 1.83876641-1.61255487i
    alternative steps
    1.22284454 × ei 7π/16 × 2 × ei (-2π/3) = 1.2228445449939 × 2 × ei (7π/16+(-2π/3)) = 2.44568909 × ei (-11π/48) = 1.83876641-1.61255487i

z20 = ((-5i)^(1/8))*(8^(1/3)) = 2.44045268+0.15995572i = 2.44568909 × ei π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei 11π/16 = -0.67937603+1.01675808i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: (-0.67937603+1.01675808i) * (-1-1.73205081i) = -0.67937602880898 * (-1) + (-0.67937602880898) * (-1.7320508075689i) + 1.0167580797323i * (-1) + 1.0167580797323i * (-1.7320508075689i) = 0.67937603+1.1767138i-1.01675808i-1.76107665i2 = 0.67937603+1.1767138i-1.01675808i+1.76107665 = 0.67937603 + 1.76107665 +i(1.1767138 - 1.01675808) = 2.44045268+0.15995572i
    alternative steps
    1.22284454 × ei 11π/16 × 2 × ei (-2π/3) = 1.2228445449939 × 2 × ei (11π/16+(-2π/3)) = 2.44568909 × ei π/48 = 2.44045268+0.15995572i

z21 = ((-5i)^(1/8))*(8^(1/3)) = 1.61255487+1.83876641i = 2.44568909 × ei 13π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei 15π/16 = -1.19934793+0.23856514i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: (-1.19934793+0.23856514i) * (-1-1.73205081i) = -1.1993479299514 * (-1) + (-1.1993479299514) * (-1.7320508075689i) + 0.23856513605852i * (-1) + 0.23856513605852i * (-1.7320508075689i) = 1.19934793+2.07733155i-0.23856514i-0.41320694i2 = 1.19934793+2.07733155i-0.23856514i+0.41320694 = 1.19934793 + 0.41320694 +i(2.07733155 - 0.23856514) = 1.61255487+1.83876641i
    alternative steps
    1.22284454 × ei 15π/16 × 2 × ei (-2π/3) = 1.2228445449939 × 2 × ei (15π/16+(-2π/3)) = 2.44568909 × ei 13π/48 = 1.61255487+1.83876641i

z22 = ((-5i)^(1/8))*(8^(1/3)) = -0.15995572+2.44045268i = 2.44568909 × ei 25π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei (-13π/16) = -1.01675808-0.67937603i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: (-1.01675808-0.67937603i) * (-1-1.73205081i) = -1.0167580797323 * (-1) + (-1.0167580797323) * (-1.7320508075689i) + (-0.67937602880898i) * (-1) + (-0.67937602880898i) * (-1.7320508075689i) = 1.01675808+1.76107665i+0.67937603i+1.1767138i2 = 1.01675808+1.76107665i+0.67937603i-1.1767138 = 1.01675808 - 1.1767138 +i(1.76107665 + 0.67937603) = -0.15995572+2.44045268i
    alternative steps
    1.22284454 × ei (-13π/16) × 2 × ei (-2π/3) = 1.2228445449939 × 2 × ei ((-13π/16)+(-2π/3)) = 2.44568909 × ei 25π/48 = -0.15995572+2.44045268i

z23 = ((-5i)^(1/8))*(8^(1/3)) = -1.83876641+1.61255487i = 2.44568909 × ei 37π/48 Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei (-9π/16) = -0.23856514-1.19934793i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: (-0.23856514-1.19934793i) * (-1-1.73205081i) = -0.23856513605852 * (-1) + (-0.23856513605852) * (-1.7320508075689i) + (-1.1993479299514i) * (-1) + (-1.1993479299514i) * (-1.7320508075689i) = 0.23856514+0.41320694i+1.19934793i+2.07733155i2 = 0.23856514+0.41320694i+1.19934793i-2.07733155 = 0.23856514 - 2.07733155 +i(0.41320694 + 1.19934793) = -1.83876641+1.61255487i
    alternative steps
    1.22284454 × ei (-9π/16) × 2 × ei (-2π/3) = 1.2228445449939 × 2 × ei ((-9π/16)+(-2π/3)) = 2.44568909 × ei 37π/48 = -1.83876641+1.61255487i

z24 = ((-5i)^(1/8))*(8^(1/3)) = -2.44045268-0.15995572i = 2.44568909 × ei (-47π/48) Calculation steps

  1. Divide: 1 / 8 = 0.125
  2. Power: (-5i) ^ 0.125 = (5 × ei (-π/2))0.125 = 50.125 × ei 0.125 × (-π/2) = 1.22284454 × ei (-5π/16) = 0.67937603-1.01675808i
  3. Divide: 1 / 3 = 0.33333333
  4. Cube root: ∛8 = -1-1.73205081i
  5. Multiple: (0.67937603-1.01675808i) * (-1-1.73205081i) = 0.67937602880898 * (-1) + 0.67937602880898 * (-1.7320508075689i) + (-1.0167580797323i) * (-1) + (-1.0167580797323i) * (-1.7320508075689i) = -0.67937603-1.1767138i+1.01675808i+1.76107665i2 = -0.67937603-1.1767138i+1.01675808i-1.76107665 = -0.67937603 - 1.76107665 +i(-1.1767138 + 1.01675808) = -2.44045268-0.15995572i
    alternative steps
    1.22284454 × ei (-5π/16) × 2 × ei (-2π/3) = 1.2228445449939 × 2 × ei ((-5π/16)+(-2π/3)) = 2.44568909 × ei (-47π/48) = -2.44045268-0.15995572i

Calculate the next expression:






This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. As imaginary unit use i or j (in electrical engineering) which satisfies basic equation i2 = −1 or j2 = −1. The calculator also provides conversion of a complex number into angle notation (phasor notation), exponential or polar coordinates (magnitude and angle). Enter expression with complex numbers like 5*(1+i)(-2-5i)^2

Complex numbers in the angle notation or phasor (polar coordinates r, θ) may you write as rLθ where r is magnitude/amplitude/radius, and θ is angle (phase) in degrees, for example, 5L65 which is same as 5*cis(65°).
Example of multiplication of two imaginary numbers in the angle/polar/phasor notation: 10L45 * 3L90.

Why next complex numbers calculator when we have WolframAlpha? Because Wolfram tool is slow and some features such as step by step are charged premium service.                  
For use in education (for example calculations alternating currents at high school) you need quick and clear complex number calculator.



Basic operations with complex numbers

We hope that work with the complex number is quite easy because you can work with imaginary unit i as a variable. And use definition i2 = -1 to simplify complex expressions. Many operations are the same as operations with two-dimensional vectors.

Addition

Very simple, add up the real parts (without i) and add up the imaginary parts (with i):
This is equal to use rule: (a+bi)+(c+di) = (a+c) + (b+d)i

(1+i) + (6-5i) = 7-4i
12 + 6-5i = 18-5i
(10-5i) + (-5+5i) = 5

Subtraction

Again very simple, subtract the real parts and subtract the imaginary parts (with i):
This is equal to use rule: (a+bi)+(c+di) = (a-c) + (b-d)i

(1+i) - (3-5i) = -2+6i
-1/2 - (6-5i) = -6.5+5i
(10-5i) - (-5+5i) = 15-10i

Multiplication

To multiply two complex number use distributive law, avoid binomials and apply i2 = -1.
This is equal to use rule: (a+bi)(c+di) = (ac-bd) + (ad+bc)i

(1+i) (3+5i) = 1*3+1*5i+i*3+i*5i = 3+5i+3i-5 = -2+8i
-1/2 * (6-5i) = -3+2.5i
(10-5i) * (-5+5i) = -25+75i

Division

Division of two complex number is based on avoid imaginary unit i from denominator. This can be done only via i2 = -1. If denominator is c+di, to make it without i (or make it real), just multiply with conjugate c-di:

(c+di)(c-di) = c2+d2

 fraction{a+bi}{c+di} = fraction{(a+bi)(c-di)}{(c+di)(c-di)} = fraction{ac+bd+i(bc-ad)}{c**2+d**2} = fraction{ac+bd}{c**2+d**2}+ fraction{bc-ad}{c**2+d**2} i ; ;

(10-5i) / (1+i) = 2.5-7.5i
-3 / (2-i) = -1.2-0.6i
6i / (4+3i) = 0.72+0.96i

Absolute value or modulus

Absolute value or modulus is distance of image of complex number from origin in plane. That use Pythagorean theorem, just as case of 2D vector. Very simple, see examples: |3+4i| = 5
|1-i| = 1.4142135623731
|6i| = 6
abs(2+5i) = 5.3851648071345

Square root

Square root of complex number (a+bi) is z, if z2 = (a+bi). Here ends simplicity. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number. If you want to find out the possible values, the easiest way is probably to go with De Moivre's formula. Here our calculator is on edge, because square root is not a well defined function on complex number. We calculate all complex roots from any number - even in expressions:

sqrt(9i) = 2.12132034+2.12132034i
sqrt(10-6i) = 3.29104116-0.91156563i
pow(-32,1/5)/5 = -0.4
pow(1+2i,1/3)*sqrt(4) = 2.43923302+0.94342254i
pow(-5i,1/8)*pow(8,1/3) = 2.39869586-0.47713027i

Square, power, complex exponentiation

Yes, our calculator can power any complex number to any integer (positive, negative), real or even complex number. In another words, we calculate 'complex number to a complex power' or 'complex number raised to a power'...
Famous example:
i**i = e**{- pi/2} ; ;
i^2 = -1
i^61 = i
(6-2i)^6 = -22528-59904i
(6-i)^4.5 = 2486.13779853-2284.55578905i
(6-5i)^(-3+32i) = 2929449.0670531-9022199.6661184i
i^i = 0.20787957635076
pow(1+i,3) = -2+2i

Functions

sqrt
Square Root of a value or expression.
sin
sine of a value or expression. Autodetect radians/degrees.
cos
cosine of a value or expression. Autodetect radians/degrees.
tan/tg
tangent of a value or expression. Autodetect radians/degrees.
exp
e (the Euler Constant) raised to the power of a value or expression
pow
Power one complex number to another integer/real/comple number
ln
The natural logarithm of a value or expression
log
The base-10 logarithm of a value or expression
abs or |1+i|
Absolute value of a value or expression
cis
is less known notation: cis(x) = cos(x)+ i sin(x); example: cis (pi/2) + 3 = 3+i
conj
conjugate of complex number - example: conj(4i+5) = 5-4i

Examples:

cube root: cuberoot(1-27i)
roots of Complex Numbers: pow(1+i,1/7)
phase, complex number angle: phase(1+i)
cis form complex numbers: 5*cis(45°)
Polar form of complex numbers: 10L60
complex conjugate: conj(4+5i)